Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13741 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| socle | C22 | C2×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 130 points:
(1,2);(3,4)(5,18)(6,13)(7,27)(8,34)(9,39)(10,42)(11,17)(12,20)(14,30)(15,29)(16,36)(19,58)(21,66)(22,68)(23,61)(24,70)(25,75)(26,57)(28,54)(31,89)(32,92)(33,79)(35,51)(37,98)(38,95)(40,59)(41,100)(43,60)(44,62)(45,81)(46,80)(47,103)(48,65)(49,73)(50,72)(52,113)(53,88)(55,85)(56,64)(63,120)(67,122)(69,118)(71,110)(74,124)(76,109)(77,83)(78,86)(82,117)(84,94)(87,102)(90,104)(91,129)(93,116)(96,114)(97,112)(99,111)(101,119)(105,121)(106,130)(107,125)(108,127)(115,123)(126,128);(3,5)(4,12)(6,19)(7,28)(8,35)(9,40)(10,43)(11,23)(13,44)(14,51)(15,54)(16,56)(17,48)(18,27)(20,30)(21,29)(22,36)(24,71)(25,76)(26,69)(31,90)(32,93)(33,82)(34,45)(37,99)(38,97)(39,46)(41,101)(42,47)(49,109)(50,110)(52,114)(53,112)(55,117)(57,105)(58,70)(59,89)(60,92)(61,79)(62,73)(63,72)(64,113)(65,88)(66,77)(67,85)(68,78)(74,125)(75,104)(80,83)(81,86)(84,96)(87,103)(91,130)(94,116)(95,107)(98,106)(100,108)(102,111)(115,120)(118,124)(119,129)(121,123)(122,126)(127,128);(3,6,11,10)(4,13,17,42)(5,19,23,43)(7,24,33,32)(8,25,38,37)(9,26,41,36)(12,44,48,47)(14,49,53,102)(15,50,55,94)(16,39,57,100)(18,58,61,60)(20,62,65,103)(21,63,67,96)(22,40,69,101)(27,70,79,92)(28,71,82,93)(29,72,85,84)(30,73,88,87)(31,74,91,86)(34,75,95,98)(35,76,97,99)(45,104,107,106)(46,105,108,56)(51,109,112,111)(52,77,115,126)(54,110,117,116)(59,118,119,68)(64,80,121,127)(66,120,122,114)(78,89,124,129)(81,90,125,130)(83,123,128,113)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420508);